## Abstract

We consider an undirected graph *G*=(*VG*,*EG*) with a set *T*⊆*VG* of terminals, and with nonnegative integer capacities *c*(*v*) and costs *a*(*v*) of nodes *v*∈*VG*. A path in *G* is a *T-path* if its ends are distinct terminals. By a *multiflow* we mean a function *F* assigning to each *T*-path *P* a nonnegative rational *weight*
*F*(*P*), and a multiflow is called *feasible* if the sum of weights of *T*-paths through each node *v* does not exceed *c*(*v*). The *value* of *F* is the sum of weights *F*(*P*), and the *cost* of *F* is the sum of *F*(*P*) times the cost of *P* w.r.t. *a*, over all *T*-paths *P*.

Generalizing known results on edge-capacitated multiflows, we show that the problem of finding a minimum cost multiflow among the feasible multiflows of maximum possible value admits *half-integer* optimal primal and dual solutions. Moreover, we devise a strongly polynomial algorithm for finding such optimal solutions.

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Babenko, M.A., Karzanov, A.V. Min-cost multiflows in node-capacitated undirected networks.
*J Comb Optim* **24**, 202–228 (2012). https://doi.org/10.1007/s10878-011-9377-3

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DOI: https://doi.org/10.1007/s10878-011-9377-3